Browsing by Author "Prof. Dr. Kashif Ali"

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Applications of Hirota Method to Nonlinear Partial Differential Equations
(2023-03-12) Muhammad Khalid; SP22-RMT-035/; LHR TP 8735; Prof. Dr. Kashif Ali
Nonlinear partial differential equations (NLPDEs) are used to simulate a wide range of physical processes in the fields of physics, engineering, mechanics, biology, and chemistry and electronics science. These NLPDE-based models provide an explanation for a wide range of physical phenomena. One special kind of NLPDEs is the nonlinear Schrodinger ¨ equation (NLSE). A prominent description in the NLSE analysis is that soliton has been a popular subject of study for nonlinear model researchers during the last two decades. The Hirota method, a popular and reliable mathematical tool for locating soliton solutions of NLPDEs in a range of disciplines, including nonlinear dynamics, mathematical physics, and engineering sciences, requires the bilinearization of NLPDEs. In this thesis, our objective is to illustrate the Exact solutions of Chafee-Infante differential equation. We obtain lump soliton, lump soliton one kink and two kink, multiwave, peri odic wave, rogue waves, periodic cross lump wave solutions, breather lump wave solutions, interaction between lump periodic kink wave. Moreover, we study on conformable frac tional extended KdV model equation. We obtain some important solutions like Homoclinic breather solution, M-shaped rational solution, M-shaped rational solution one kink and two kink, multiwave, periodic cross rational solutions and periodic cross kink wave solutions.
Mathematical Modeling and Qualitative Analysis of Dynamical Systems
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Areej Abbas; CIIT/SP23-RMT-004/LHR; Prof. Dr. Kashif Ali; LHR TP 9590
Many harmful and dangerous incidents have occurred throughout human history, restrict- ing people’s way of life and killing millions of people worldwide. It is helpful for analyzing a variety of natural disasters using mathematical methods and creating strategies for these pandemic. This perspective uses differential equations and speciﬁc epidemiological con- straints to develop mathematical models. Mathematical modeling have many extensions as epidemiological models, economic models, climate models, biological models, engineer- ing models etc. An important tool for understanding the dynamics of infectious illnesses is the mathematical modeling of epidemiological systems. These models helps in the eval- uation of control strategies, making projections of disease transmission, and the oversight of public health initiatives. The simplest epidemiological dynamical model introduced by Kermack and McKendrick in 1927 is the SIR model. This compartmental model offers basic insights into the spread of disease and possible points of intervention by describing how individuals go through stages of susceptibility, infection and recovery. The SIR model has multiple modiﬁcations such as ebola epidemic SEIR model by Raﬁq et al. that is clas- siﬁed into susceptible, exposed, infected and recovered. This model tells us that how the susceptible individuals contact with infected people and becomes infected after exposed of ebola symptoms then moves from infected class to recovery class with a speciﬁc recov- ery rate. Another example is corruption dynamics with optimal control strategies (SCJH) model by Duressa and Legesse is classiﬁed as S (susceptible), C (corrupted), J (jailed) and H (honest). In this model we deal with corruption as an infectious disease and we use different strategies for overcome corruption dynamics. After analyzing the epidemio- logical dynamical model, we ﬁnd reproductive number (Ro) which denotes the number of secondary infections caused by an infected individual in a population who is completely susceptible, is crucial to these models by using the next generation matrix method with the help of transmission and disease matric. If Ro > 1, then the disease invade and prolif- erate. If Ro < 1, then disease will die out. Understanding the behavior of the disease at ix both small and large scales requires an understanding of local and global stability analy- ses. Local stability is concerned with how solutions behave close to an equilibrium point whereas global stability takes into account the system’s general behavior under any ini- tial conditions. An endemic or disease-free equilibrium can be established and sustained with the help of these analyses. The simulations that numerical analysis offers can han- dle complex models and real-world data which is a complement to theoretical approaches. Researchers can investigate real world problems that are analytically unsolvable, analyze them and through numerical simulations predict the effects of various solutions. In this study, epidemiological modeling is compiled with an emphasis on numerical simulations, stability evaluations and reproductive number calculation. In infectious disease control by offering a thorough review, this study intends to improve knowledge and implementation of mathematical models. The present study focuses on the dynamical analysis of different dynamical models. Ini- tially, the dynamical analysis of epidemiological models will be discussed by evaluating the reproductive number and disease-free and endemic equilibrium points to check the sta- bility of the system. In addition, modeling in a variety of scientiﬁc domains, including chemistry and physics, will be covered. Secondly, we will apply various numerical tech- niques to the models, such as the non-standard ﬁnite difference approach, and compare the outcomes with the model’s theoretical predictions. Inﬂuenza virus, an infection which be- comes a global threat, causing seasonal outbreaks and pandemic. Current vaccines against inﬂuenza A and B viruses are vulnerable to epidemic strains that are poorly matched by the vaccine. A vaccination that is less sensitive to virus antigenic(virus’s surface proteins) evolution would be a signiﬁcant improvement. The present study introduces a nonlinear differential model by including hospitalized compartment. In this paper we have to prove the non-negativity and boundedness of the model where non-negativity tells us that the parameters in the model cannot take negative values for example population cannot be neg- ative means it represents reality in a logical way. When variables in a model are bounded, it means that they have upper bounds and are not able to increase inﬁnitely. This is a reﬂection of real-world constraints such as ﬁnite resources, physical limitations, or envi- ronmental conditions. Also ﬁnd the sensitivity analysis which describes how changes in a model’s input parameters affect its output and demonstrates which parameters have the greatest inﬂuence on the model’s behavior or output, and which have little to no impact. x We have perform the stability analysis to explore the inﬂuential parameters on the growth of inﬂuenza virus by calculating the inﬂuenza free equilibrium (IFE) and inﬂuenza en- demic equilibrium (IEE) points, and computes the fundamental reproduction number of inﬂuenza (Rin f .) by applying next generation matrix method that indicates how many new infections each infected individual will typically generate in a population that is fully sus- ceptible. We show that the IFE is locally asymptotically stable (LAS) if Rin f . < 1 implies that disease will not spread widely or will eventually die out, as each infected individual is, on average, causing less than one new infection. When the Rin f . > 1 , the IEE is sta- ble locally in an asymptotic manner means the infection will grow and the population will not return to the disease-free state. We generally proves the global stability analysis with help of Lyapunov function ensures that the system will respond in a predictable manner over time, either moving in the direction of the disease’s eradication or stabilising in a state where it is constantly present but under control. We verify these theoretical results through numerical simulation by applying non-standardized ﬁnite difference (NSFD) scheme
Mostar Invariants of Different Families of Graphs
(Library Information Services, CUI Lahore, 2021) Noor-ul-Huda; SP21-RMT-015/; Prof. Dr. Kashif Ali
Mostar invariant is an advance bond additive invariant that was presented in the near past. We obtain results for different variants of weighted Moster indices. Furthermore, we de- termine the numerical expressions for different versions of weighted plus and product PI indices for chemical structures such as Melem chain nanostructure, Poly-methyl methacry- late network PMMA and SIO
Soliton Solutions to Electromagnetic Model via Two Integration Schemes
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Syed Zaheer Abbas; CIIT/SP21-BSM-005/LHR; Prof. Dr. Kashif Ali; LHR TP 9879
Solitons are the special types of solitary waves who does not change their shape and speed during propagation. When a light pulse in space or time travels in some host medium, it does not dispersive or diffract is called optical Solitons. Optical solitons are produced due to balance between group velocity dispersion and self-phase modulation. In this thesis, we have studied the electromagnetic model with two integration schemes namely the enhanced algebraic method and Riccati equation method. Our purpose is to attain bell-type, kink and anti kink type, singular solitons solutions for various forms of NLEEs. We’ll also obtain Weierstrass elliptic functions, Jacobi elliptic functions in terms of hyperbolic, trigonometric and rational functions. We have present the solutions graphically in various dimensions like 3D, 2D and contour.
Study of Nonlocal Differential Equations
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Syeda Zurriat Fatima Zaidi (fa23-rmt-039); Prof. Dr. Kashif Ali; LHR TP 9784
This thesis examines the theory, classification, and solution behavior of second-, third, and fourth-order nonlocal differential equations. Nonlocal differential equations, which are distinguished by the occurrence of terms involving integration or dependence on values of the unknown function over an interval or the whole domain, have attracted considerable attention because of their applications in mathematical physics, control theory, and continuum mechanics. We starts with a thorough examination of second-order nonlocal linear differential equations, their overall form, solution techniques, and stability behavior. This is then generalized to third-order nonlocal equations, where complexity increases through derivatives of order greater than one and increased functional dependence. Lastly, a complete framework is established for fourth-order nonlocal differential equations, giving homogeneous and nonhomogeneous cases with different nonlocal kernels. Every order is analyzed with example cases motivated by current literature, and closed-form solutions when possible are obtained. Existence, uniqueness, and qualitative solution behavior are also considered. The thesis adds to the general comprehension of higher-order nonlocal operators and gives one a single way of solving and analyzing these equations systematically
Use of Artificial Intelligence in Group Representation Theory
(Library Information Services, CUI Lahore, 2022) Ali Haidar; CIIT/FA21-RMT-103/LHR; Prof. Dr. Kashif Ali
In this thesis, we aimed to explore the intersection between artificial intelligence and repre- sentation theory. We began by providing a comprehensive overview of the necessary back- ground material, including artificial intelligence and machine learning, neural networks, and graph neural networks, as well as representation theory. To further illustrate the prac- tical applications of these techniques, we analyzed a recent paper [9] that leveraged graph neural networks to improve the state-of-the-art in a specific problem in representation the- ory. We replicated their code and results, with the goal of gaining a deeper understanding of the techniques they employed and the insights they gleaned.